Processos estocásticos e difusão anômala em sistemas complexos

Abstract

Stochastic processes play an important role in the dynamics of complex systems. Such systems are compound by several elements which may interact nonlinearly, leading to a non-trivial global behavior; and/or show high structural complexity. In this context, the present thesis is dedicated to the study of diffusive processes in complex systems, mainly focused on anomalous diffusion. Our study begins by contextualizing diffusive processes in the study of these systems, in order to relate the mechanisms of anomalous diffusion to the nature of the interactions and to the structural heterogeneities. Thus, we discuss some generalizations of the conjectures of usual diffusion proposed to model complex system, i.e., concepts and mathematical methods give by: i) continuous time random walk; ii) fractional diffusion equation; and iii) generalized Langevin equation. Following, we investigate some extensions of the diffusion equation with geometrical constraints, called comb model. In particular, we discuss the relation between the spreading of the system with the presence of external forces, and with the presence of the backbone term. We also analyze the first passage time and the survival probability of the comb model. By means of time dependent analytical solutions, obtained by using integral transform and the Green's Functions approach, we show how geometrical constraints and memory effects(fractional derivatives), can lead to a rich class of anomalous diffusive behaviors. Finally, we present our general conclusions.